Workshop Session 1

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---
class: left, bg_aiu

<BR/><BR/><BR/><BR/><BR/><BR/>
## ECN339 Finance Data Workshop

<BR/>

### Introduction +<br/>Finance Terms, R, RStudio, Nikkei Needs
#### [Clinton Watkins](https://www.clintonwatkins.com)<BR/>Global Business Program, Akita International University
#### Winter 2022

---

## Intro to the course

### Objectives

1.	Develop the ability to conduct empirical analyses and interpret results.

2.	Gain an understanding of stock market data and selected financial models.

3.	Gain familiarity with the methodology of empirical research and coding in R.<p/>

### Useful links

- [ECN339 AIMS Page](https://aims.aiu.ac.jp/course/view.php?id=4974)

- [ECN339 Slack Workspace invite](https://join.slack.com/t/ecn339w2022/shared_invite/zt-100i0j7fr-SZ6I9NWl2JToVelE_IRPOg)

- [Setting up R and RStudio](https://www.clintonwatkins.com/courses/fd/fdw_r_intro_nb_20211218.nb.html). Hopefully you have already done this.<p/>

---

## Syllabus

- [Online syllabus](https://www.clintonwatkins.com/courses/fd/syllabus_websitev_20211119.html) on my website and on AIMS.<p/>

---

## Topics

> We will investigate and model the cross-section of Japanese stock returns with R and RStudio using Nikkei Needs financial data on Tokyo Stock Exchange listed firms.<sup>1<sup/>

1. Introduction
    - R and RStudio, basics of linear algebra, OLS estimator.
    - Finance terms, Nikkei Needs Japanes stocks database.

2. Topic 1: Factor Models
    - Market model, Fama-French models, principle components.
    
3. Topic 2: Fama Macbeth model for panel data
    - Estimating risk premia.
    
4. Topic 3: Bankruptcy Risk and quantile portfolios
    - Z-score, O-score

5. Wrap-up and Quiz.

.footnote[[1] We will approach the course topics flexibly and they may be adjusted depending on our pace through the course.]
---

## R and RStudio

- R is a powerful language suited to statistical analysis, graphics and many other applications.  
- RStudio provides a convenient interface for using R.

- Both are free. We will use R and RStudio for this course.

- Hopefully you have already downloaded and set up both using the [guide on my website](https://www.clintonwatkins.com/courses/fd/fdw_r_intro_nb_20211218.nb.html).

- [R statistical programming language <img src="data:image/png;base64,#assets/Rlogo.png" height="20px" />](https://www.r-project.org/).
    
    - [RStudio <img src="data:image/png;base64,#assets/rstudio-icon.png" height="20px"/>](https://www.rstudio.com/) and [RStudio Cloud](https://rstudio.cloud/).
    
<center><img src="data:image/png;base64,#assets/rstudio1.jpg" height="250px"/></center>

???

Please download and install this software for the course.

---
class: center middle

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
Important in this workshop course:

Enter and execute the code for the analyses in RStudio as we go through them.

If you encounter an error, do not hesitate to ask me for help.
]

---

## If you haven't set up RStudio yet...

- Use [RStudio Cloud](https://rstudio.cloud/) for this session.

- Make a free [RStudio Cloud](https://rstudio.cloud/) account now (provides limited free monthly computing hours, likely not enough for this course).
    
    - Create a new project for your work in this course.
    
    - You will probably need to load some packages.
    
    - Do not forget to download any files you create for use later on your own computer.
    
- Set up R and RStudio on your computer by the next class and do the pre-assignment.

---

## R and RStudio [set-up guide](https://www.clintonwatkins.com/courses/fd/fdw_r_intro_nb_20211218.nb.html)

---

## Resources for R

.pull-left[
### Online guides

- A comprehensive [introduction to R](https://cran.r-project.org/doc/manuals/r-release/R-intro.pdf).

- A textbook [Introduction to Econometrics with R](https://www.econometrics-with-r.org/).

- [Documentation and guides for various applications contributed by R users](https://cran.r-project.org/other-docs.html#english).

- [Massimiliano's R Page on GitHub](https://github.com/massimilianoj/R_beginners) contains [R for Beginners](https://github.com/massimilianoj/R_beginners/blob/master/R_beginners.pdf), used in my econometrics, finance and macroeconomics classes in Kobe-U.

- [Massimiliano](https://link.springer.com/book/10.1007/978-3-030-34529-7) worked for me as a TA and RA in Kobe University. We are working on a book about analysing foreign exchange markets using R.

]

.pull-right[

### CRAN Task Views

- The [Comprehensive R Archive Network](https://cran.r-project.org/) (CRAN) Task Views provide information about packages that can be used with R.

- Packages extend the functionality of R. They provide routines for various types of data manipulation, analysis, graphics and visualisation, econometric models, and many more putposes.

- Packages are constantly being developed and updated by R users.

- The most relevant Task View for this course is [Empirical Finance](https://CRAN.R-project.org/view=Finance)

]

---

## Resources for R

.pull-left[

### User community
- R is an open source language developed by users and has a huge online community.

- A good place to look for Q&A and help is [stackoverflow](https://stackoverflow.com/questions/tagged/r).

### Google is your friend

- If you are stuck, [Google search](https://www.google.com/) for information that will help you answer your question.

- [R Seek <img src="data:image/png;base64,#assets/Rlogoseek.jpg" height="20px"/>](https://rseek.org/) is a search engine for R.

]

.pull-right[
### R-Bloggers

- [<img src="data:image/png;base64,#assets/R_02.jpg" height="20px"/> R-Bloggers](https://www.r-bloggers.com/) contains many useful posts and [Tutorials](https://www.r-bloggers.com/how-to-learn-r-2/) on how to use R for a wide variety of applications (not just finance).

### Books

- There are many good books on using R.

- Tsay (2014), Ang (2021) and Bennett and Hugen (2016) for finance.
    - Kleiber and Zeileis (2008) and Pfaff (2008) for econometrics.
    - Matloff (2011) is good for general programming in R.

]

---

```r
library(rfishdraw)
file <- system.file("extdata",package = "rfishdraw")
dat <- readRDS(paste0(file,"/","output_json.rds"))
for (i in seq(length(dat))) {dat[[i]] <- cbind(dat[[i]],i)}
plot(NA, xlim = c(0,520),ylim = c(-320,0),axes = F,xlab = " ",ylab = " ")
for(i in seq(length(dat))) {lines(x = dat[[i]][,1],
                                  y = -dat[[i]][,2], lwd=2, col = "blue")}
```

---

## A map of ...

```r
library(leaflet)
leaflet() %>% addTiles() %>% 
  setView(140.19838192985657, 39.62662109562309, zoom = 17)
```

<div id="htmlwidget-5aec1ee595af3dcafaed" style="width:100%;height:424.8px;" class="leaflet html-widget"></div>
<script type="application/json" data-for="htmlwidget-5aec1ee595af3dcafaed">{"x":{"options":{"crs":{"crsClass":"L.CRS.EPSG3857","code":null,"proj4def":null,"projectedBounds":null,"options":{}}},"calls":[{"method":"addTiles","args":["//{s}.tile.openstreetmap.org/{z}/{x}/{y}.png",null,null,{"minZoom":0,"maxZoom":18,"tileSize":256,"subdomains":"abc","errorTileUrl":"","tms":false,"noWrap":false,"zoomOffset":0,"zoomReverse":false,"opacity":1,"zIndex":1,"detectRetina":false,"attribution":"© <a href=\"http://openstreetmap.org\">OpenStreetMap<\/a> contributors, <a href=\"http://creativecommons.org/licenses/by-sa/2.0/\">CC-BY-SA<\/a>"}]}],"setView":[[39.6266210956231,140.198381929857],17,[]]},"evals":[],"jsHooks":[]}</script>

---
class: middle

### [Conway's Game of Life](https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life) [(R-Bloggers)](https://www.r-bloggers.com/2012/11/fast-conways-game-of-life-in-r/)

.pull-left-45[

<BR/>

]

.pull-right-55[

```r
library(caTools)
side=500
steps=300
X <- matrix(nrow=side, ncol=side)
X[] <- rbinom(side^2,1,0.4)
storage <- array(0, c(side, side, steps))
# the simulation                                             
for (i in 1:steps)
{
  allW = cbind( rep(0,side), X[,-side])
  allNW = rbind(rep(0,side),
          cbind(rep(0,side-1),X[-side,-side]))
  allN = rbind(rep(0,side),X[-side,])
  allNE = rbind(rep(0,side),
          cbind(X[-side,-1],rep(0,side-1)))
  allE =  cbind(X[,-1],rep(0,side))
  allSE = rbind(cbind(X[-1,-1],
          rep(0,side-1)),rep(0,side))
  allS = rbind(X[-1,],rep(0,side))
  allSW = rbind(cbind(rep(0,side-1),
          X[-1,-side]),rep(0,side))
  X2 <- allW + allNW + allN + allNE +
    allE + allSE + allS + allSW
  X3 <- X
  X3[X==0 & X2==3] <- 1
  X3[X==1 & X2<2] <- 0
  X3[X==1 & X2>3] <- 0
  X <- X3
  storage[,,i] <- X2
}
storage <- storage/max(storage)
write.gif(storage, "conway.gif", 
          col="jet", delay=5)
```

]

---

### Build websites using R blogdown

---

### Shiny apps built using R

---

### Shiny app examples...

---

### Top 8 things to know about R

1. R is an *object oriented language*. Everything in R is an object with particular properties that can interact with other objects in certain ways. Understanding the different types of objects is key to using them. E.g. vector, matrix, data frame, list, function.

2. The *assignment operator* is used to assign values to an object.
    
    ```r
    a <- 2
    b <- 3
    a+b
    ```
    
    ```
    ## [1] 5
    ```

3. *Packages* extend the functionality of base R. Packages contain code (functions), data and documents. To use a package it must be installed (once) and then loaded each session when you want to use it. Find information about packages from Task Views or internet search.
    
    ```r
    install.packages("leaflet")
    library(leaflet)
    ```

4. When you have an *error* executing your code, 99% of the time it will be because (i) you mistyped the code or (ii) you have not loaded a package required for a function to work.

---

### Top 8 things to know about R

<ol start=5>
  <li> The *working directory* is where R will look for and save files.

```r
getwd()
```

```
## [1] "/Users/clinton/Dropbox/syncd_r_data/FDW/lectures/1/lecture"
```

```r
setwd("/Users/clinton/Dropbox/syncd_r_data/FDW/preparation")
```
  </li>
  
  <li> Lots of documentation and help is available within R. Try:

```r
?seq
help(seq)
example(seq)
help(package - AER)
help (topic = "Arithmetic")
help.search("regression")
??regression
```
  </li><p/>
  
  <li> *Reproducible research:* do your work in an R notebook, including explanation, code and output. This way your work can be explained to tohers and will be reproducible later, even when you may have forgotten exactly what you did. 
  </li><BR/>
  
  <li> Practice *writing code* as much as possible. It will be frustrating at first, but your skills will improve rapidly if you keep at it.
  </li>
<ol>

---
class: center middle

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
The only way to write good code is to write tons of bad code first. 
Feeling shame about bad code stops you from getting to good code.

.tr[
— Hadley Wickham
]]

---

### R Notebooks

.pull-left[

- I've uploaded an R Notebook template to AIMS showing many of the useful features.

- Download the template and use during the class and for your homework.

- Open a new Notebook from the menu at the top-right of the RStudio toolbar or write in the template I provided.

- A Notebook consists of the YAML header, text, code chunks and output.

- Cut, paste and modify examples from the template.

- Saving the .Rmd file produces or updates the .html file Notebook.

- Remember to run all code (menu: Run All) before saving your Notebook.

]

.pull-right[

***

]

---

### Basics of linear algebra in R

- Knowing some basics of linear algebra will help you understand both how your R code works and the theory behind the financial econometric models that we will cover.

- A scalar object that represents a single value (numeric, character, logical).

- Assigning a random scalar:
    
    ```r
    a <- sample(1:10, 1)
    a
    ```
    
    ```
    ## [1] 5
    ```

- A vector is a sequence of elements of the same data type (numeric, character, logical).

`$$x = \begin{bmatrix}
x_1\\
x_2\\
\vdots\\
x_n
\end{bmatrix}$$`

- A vector may be a row vector or column vector (like `$x$` above).

- Vector arithmetic is done element by element, or member-wise.

---

### Vector arithmetic

.pull-left[

- Create two vectors by concatenation:
    
    ```r
    a <- c(1, 3, 5, 7)
    b <- c(1, 2, 4, 8)
    ```

- Arithmetic with scalars:
    
    ```r
    c <- 2 * a + 3
    c
    ```
    
    ```
    ## [1]  5  9 13 17
    ```

- Addition:
    
    ```r
    a + b
    ```
    
    ```
    ## [1]  2  5  9 15
    ```

- Subtraction:
    
    ```r
    a - b
    ```
    
    ```
    ## [1]  0  1  1 -1
    ```
]

.pull-right[
- Multiplication:
    
    ```r
    a * b
    ```
    
    ```
    ## [1]  1  6 20 56
    ```

- Division:
    
    ```r
    a / b
    ```
    
    ```
    ## [1] 1.000 1.500 1.250 0.875
    ```

- What is the length of the vector?
    
    ```r
    length(a)
    ```
    
    ```
    ## [1] 4
    ```

- Recycling rule: where vectors are of unequal length, the shorter is recycled to match the length of the longer.
    
    ```r
    d <- c(1, 3, 5, 7, 9, 11, 13, 15)
    a + d
    ```
    
    ```
    ## [1]  2  6 10 14 10 14 18 22
    ```

]

---

### Vectors

.pull-left[

- Create another vector:
    
    ```r
    x <- c(1.8, 3.14, 4, 88.169, 13)
    x
    ```
    
    ```
    ## [1]  1.800  3.140  4.000 88.169 13.000
    ```

- Subsetting the first and fourth elements using square brackets:
    
    ```r
    x[c(1,4)]
    ```
    
    ```
    ## [1]  1.800 88.169
    ```

- Or subset by excluding elements:
    
    ```r
    x[-c(2,3,5)]
    ```
    
    ```
    ## [1]  1.800 88.169
    ```

- We can use various operators on vectors, e.g.:
    
    ```r
    round(log(x),3)
    ```
    
    ```
    ## [1] 0.588 1.144 1.386 4.479 2.565
    ```
    
]

.pull-right[ 
- Joining vectors:
    
    ```r
    e <- c(a,b)
    e
    ```
    
    ```
    ## [1] 1 3 5 7 1 2 4 8
    ```
  
- A vector or matrix can be numeric, logical, character. All elements must be the same type.
    
    ```r
    mode(x)
    ```
    
    ```
    ## [1] "numeric"
    ```
    
    ```r
    p <- x >= 4
    p
    ```
    
    ```
    ## [1] FALSE FALSE  TRUE  TRUE  TRUE
    ```
    
    ```r
    mode(p)
    ```
    
    ```
    ## [1] "logical"
    ```
    
    ```r
    q <- c("a", "b", "c")
    mode(q)
    ```
    
    ```
    ## [1] "character"
    ```
]

---

### Sequences and patterned vectors

.pull-left[

- Generate a sequence:
    
    ```r
    5:1
    ```
    
    ```
    ## [1] 5 4 3 2 1
    ```

- Using the seq function:
    
    ```r
    seq(1,5,1)
    ```
    
    ```
    ## [1] 1 2 3 4 5
    ```

- Include the sequence in vector arithmetic:
    
    ```r
    5:1*x+1:5
    ```
    
    ```
    ## [1]  10.000  14.560  15.000 180.338  18.000
    ```

]

.pull-right[

- Various functions can be used to create patterned vectors.
    
    ```r
    ones <- rep(1,10)
    ones
    ```
    
    ```
    ##  [1] 1 1 1 1 1 1 1 1 1 1
    ```
    
    ```r
    even <- seq(from=2, to=20, by=2)
    even
    ```
    
    ```
    ##  [1]  2  4  6  8 10 12 14 16 18 20
    ```
    
    ```r
    trend <- 1995:2017
    trend
    ```
    
    ```
    ##  [1] 1995 1996 1997 1998 1999 2000 2001 2002
    ##  [9] 2003 2004 2005 2006 2007 2008 2009 2010
    ## [17] 2011 2012 2013 2014 2015 2016 2017
    ```
]

***

- Combine two vectors:
    
    ```r
    c(ones,even)
    ```
    
    ```
    ##  [1]  1  1  1  1  1  1  1  1  1  1  2  4  6  8 10 12 14 16 18 20
    ```

---

### Matrices

.pull-left[

- A matrix is a two dimensional array of elements.

<BR/>

`$$\underset{2\times 3}A = \begin{bmatrix}
1 & 3 & 5\\
2 & 4 & 6
\end{bmatrix}$$`

<BR/>

`$$\underset{2\times 2}B = \begin{bmatrix}
2 & 6\\
4 & 8
\end{bmatrix}$$`

<BR/>

- The dimension of a matrix are found by counting the number of rows and columns.

- A is a 2 by 3 matrix.
    
    - B is a square 2 by 2 matrix.
    
]

.pull-right[

- Creating a matrix in R:
    
    ```r
    A <- matrix(1:6, nrow=2, byrow = FALSE)
    A
    ```
    
    ```
    ##      [,1] [,2] [,3]
    ## [1,]    1    3    5
    ## [2,]    2    4    6
    ```
    
    ```r
    B <- matrix(c(2,4,6,8), ncol=2, byrow = FALSE)
    B
    ```
    
    ```
    ##      [,1] [,2]
    ## [1,]    2    6
    ## [2,]    4    8
    ```
    
- Dimensions:
    
    ```r
    dim(A)
    ```
    
    ```
    ## [1] 2 3
    ```
    
    ```r
    dim(B)
    ```
    
    ```
    ## [1] 2 2
    ```
    
]

---

### Transpose of a matrix

.pull-left[

-  The transpose of a matrix takes each row and makes it a column. That is, it switches the row and column indices.

`$$\underset{2\times 3}A = \begin{bmatrix}
1 & 3 & 5\\
2 & 4 & 6
\end{bmatrix}$$`

- The transpose of `$A$` is `$A^{\prime}$`:

`$$\underset{3\times 2}A^{\prime} = \begin{bmatrix}
1 & 2\\
3 & 4\\
5 & 6
\end{bmatrix}$$`

]

.pull-right[

- Recall matrix A:
    
    ```r
    A
    ```
    
    ```
    ##      [,1] [,2] [,3]
    ## [1,]    1    3    5
    ## [2,]    2    4    6
    ```

- Transpose of matrix A:
    
    ```r
    t(A)
    ```
    
    ```
    ##      [,1] [,2]
    ## [1,]    1    2
    ## [2,]    3    4
    ## [3,]    5    6
    ```

]

---

### Subsetting and adding to a matrix

.pull-left[
- Subset a matrix the same way that we subset a vector.
    
    ```r
    A
    ```
    
    ```
    ##      [,1] [,2] [,3]
    ## [1,]    1    3    5
    ## [2,]    2    4    6
    ```
    
    ```r
    A[2,3]
    ```
    
    ```
    ## [1] 6
    ```

- To subset rows 1 and 2, and columns 1 and 3:
    
    ```r
    A1 <- A[1:2,c(1,3)]
    A1
    ```
    
    ```
    ##      [,1] [,2]
    ## [1,]    1    5
    ## [2,]    2    6
    ```
    
    ```r
    A[,-2]
    ```
    
    ```
    ##      [,1] [,2]
    ## [1,]    1    5
    ## [2,]    2    6
    ```

]
 
.pull-right[
- Add a row to a matrix:
    
    ```r
    rbind(A, c(10,20, 30))
    ```
    
    ```
    ##      [,1] [,2] [,3]
    ## [1,]    1    3    5
    ## [2,]    2    4    6
    ## [3,]   10   20   30
    ```
    
- Add a column to a matrix:
    
    ```r
    cbind(A, c(10,20))
    ```
    
    ```
    ##      [,1] [,2] [,3] [,4]
    ## [1,]    1    3    5   10
    ## [2,]    2    4    6   20
    ```

- Or join two matrices:
    
    ```r
    cbind(A, A1)
    ```
    
    ```
    ##      [,1] [,2] [,3] [,4] [,5]
    ## [1,]    1    3    5    1    5
    ## [2,]    2    4    6    2    6
    ```
]

---

### Element-wise arithmetic

- Addition and subtraction of matrices is always done element by element.

- Multiplication and division may be done element wise. Multilplication of two matrices of equal dimensions is called the Hadamard Product.

- A conformable matrix is a matrix with the correct dimensions for a certain matrix operation.

- Conformable matrices for element-wise operations have the same dimensions.
    
- For example, addition of the two matrices `$A$` and `$B$` is performed element-wise:

<BR/>

`$$\underset{2\times 3} A = \begin{bmatrix}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}
\end{bmatrix} \quad
\underset{2\times 3} B = \begin{bmatrix}
b_{11} & b_{12} & b_{13}\\
b_{21} & b_{22} & b_{23}
\end{bmatrix}\\[24pt]
A + B = \begin{bmatrix}
a_{11}+b_{11} & a_{12}+b_{12} & a_{13}+b_{13}\\
a_{21}+b_{21} & a_{22}+b_{22} & a_{23}+b_{23}
\end{bmatrix}\\$$`

- Subtraction is performed in the same way.

---

### Element-wise arithmetic

.pull-left[
- Consider matrices A and B:
    
    ```r
    A1
    ```
    
    ```
    ##      [,1] [,2]
    ## [1,]    1    5
    ## [2,]    2    6
    ```
    
    ```r
    B
    ```
    
    ```
    ##      [,1] [,2]
    ## [1,]    2    6
    ## [2,]    4    8
    ```
    
- Element by element addition:
    
    ```r
    A1 + B
    ```
    
    ```
    ##      [,1] [,2]
    ## [1,]    3   11
    ## [2,]    6   14
    ```

- Element by element subtraction:
    
    ```r
    A1 - B
    ```
    
    ```
    ##      [,1] [,2]
    ## [1,]   -1   -1
    ## [2,]   -2   -2
    ```
]

.pull-right[
- Element by element multiplication:
    
    ```r
    A1 * B
    ```
    
    ```
    ##      [,1] [,2]
    ## [1,]    2   30
    ## [2,]    8   48
    ```

- Element by element division:
    
    ```r
    A1 / B
    ```
    
    ```
    ##      [,1]      [,2]
    ## [1,]  0.5 0.8333333
    ## [2,]  0.5 0.7500000
    ```

- We can also multiply a matrix by a scalar:
    
    ```r
    3 * A1
    ```
    
    ```
    ##      [,1] [,2]
    ## [1,]    3   15
    ## [2,]    6   18
    ```
]

---

### Vector dot product

- The dot product of a `$n \times 1$` vector `$a$` with another `$n \times 1$` vector `$b$` is:

`$$a^{\prime}b = \begin{bmatrix} a_{1} & a_{2} & \dots & a_{n} \end{bmatrix}\begin{bmatrix}
b_{1}\\
b_{2}\\
\vdots\\
b_{n}
\end{bmatrix} =
a_{1}\times b_{1} + a_{2}\times b_{2} + \dots + a_{n}\times b_{n}$$`

- The dot product results in a scalar.

- Note that `$a^{\prime}b$` equals `$b^{\prime}a$`.

- Two vectors are said to be orthogonal if their dot product equals zero.

- Example:

`$$\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}\begin{bmatrix}
4\\
5\\
6
\end{bmatrix} =
1\times 4 + 2\times 5 + 3\times 6 = 4 + 10 + 18 = 32$$`

---

### Matrix multiplication

- Two matrices `$A$` and `$B$` can be multiplied if they are conformable for multiplication.

- To be conformable, the number of columns of A must equal the number of rows of B.
    
    - If `$A$` is `$r\times n$` and `$B$` is `$n\times c$` then `$AB$` will be `$r\times c$`.
    
    - Element `$jk$` of the product AB is equal to the dot product between row `$j$` of `$A$` and column `$k$` of `$B$`. For example:
    
`$$\underset{2\times 2} A = \begin{bmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{bmatrix} \quad
\underset{2\times 3} B = \begin{bmatrix}
b_{11} & b_{12} & b_{13}\\
b_{21} & b_{22} & b_{23}
\end{bmatrix}\\[24pt]
\underset{2\times 3}{AB} = \begin{bmatrix}
a_{11}\times b_{11}+a_{12}\times b_{21} & a_{11}\times b_{12}+a_{12}\times b_{22} & a_{11}\times b_{13}+a_{12}\times b_{23}\\
a_{21}\times b_{11}+a_{22}\times b_{21} & a_{21}\times b_{12}+a_{22}\times b_{22} & a_{21}\times b_{13}+a_{22}\times b_{23}
\end{bmatrix}\\$$`
    
- Note that in general, the order of matrix multiplication matters, `$AB\neq BA$`.

- In the example above, we say `$A$` is post-multiplied by `$B$` or `$B$` is pre-multiplied by `$A$`.
    
- The transpose of a matrix product is equal to the product of the transposed matrices in reverse order, `$(AB)^{\prime} = B^{\prime}A^{\prime}$`.

---

### Matrix multiplication

.pull-left[
- Recall:
    
    ```r
    A
    ```
    
    ```
    ##      [,1] [,2] [,3]
    ## [1,]    1    3    5
    ## [2,]    2    4    6
    ```
    
    ```r
    A1
    ```
    
    ```
    ##      [,1] [,2]
    ## [1,]    1    5
    ## [2,]    2    6
    ```
    
    ```r
    B
    ```
    
    ```
    ##      [,1] [,2]
    ## [1,]    2    6
    ## [2,]    4    8
    ```
]

.pull-right[
- `$A1\times B$`
    
    ```r
    A1 %*% B
    ```
    
    ```
    ##      [,1] [,2]
    ## [1,]   22   46
    ## [2,]   28   60
    ```

- `$A\times B$`
    
    ```r
    A %*% B
    ```
    
    ```
    ## Error in A %*% B: non-conformable arguments
    ```

- `$A^{\prime}\times B$`
    
    ```r
    t(A) %*% B
    ```
    
    ```
    ##      [,1] [,2]
    ## [1,]   10   22
    ## [2,]   22   50
    ## [3,]   34   78
    ```

]

---

### Trace and determinant

.pull-left[

- The trace of a square matrix is the sum of the diagonal elements.

`$$\underset{2\times 2} A = \begin{bmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{bmatrix}$$`
`$$tr(A) = a_{11} + a_{22}$$`

- For example:
    
    ```r
    A1
    ```
    
    ```
    ##      [,1] [,2]
    ## [1,]    1    5
    ## [2,]    2    6
    ```
    
    ```r
    fBasics::tr(A1)
    ```
    
    ```
    ## [1] 7
    ```
    
    ```r
    sum(diag(A1))
    ```
    
    ```
    ## [1] 7
    ```

]

.pull-right[
- The determinant of a square matrix characterises some of the properties of the matrix.

`$$det(A) = \begin{vmatrix} A \end{vmatrix} = a_{11}\times a_{22} - a_{12}\times a_{21}$$`

- For a `$3\times 3$` matrix:

`$$E = \begin{bmatrix} 
a & b & c\\
d & e & f\\
g & h & i
\end{bmatrix}\\[12pt]
\begin{vmatrix} E \end{vmatrix} = a \begin{vmatrix} 
e & f\\
h & i\end{vmatrix} + b \begin{vmatrix} 
d & f\\
g & i\end{vmatrix} + c \begin{vmatrix} 
d & e\\
g & h\end{vmatrix}$$`

```r
E<-matrix(c(7,9,4,2,5,6,1,3,8),nrow = 3)
E
```

```
##      [,1] [,2] [,3]
## [1,]    7    2    1
## [2,]    9    5    3
## [3,]    4    6    8
```

```r
det(E)
```

```
## [1] 68
```

]

---

### Inverse of a Matrix

.pull-left[
- The inverse of a `$2\times 2$` matrix is:
`$$A^{-1} = 
\frac{1}{\begin{vmatrix}A\end{vmatrix}}
\begin{bmatrix}
a_{22} & -a_{12}\\
-a_{21} & a_{11}
\end{bmatrix}$$`

- In general, only square matrices with a non-zero determinant are invertible.
    
    ```r
    A1
    ```
    
    ```
    ##      [,1] [,2]
    ## [1,]    1    5
    ## [2,]    2    6
    ```
    
    ```r
    solve(A1)
    ```
    
    ```
    ##      [,1]  [,2]
    ## [1,] -1.5  1.25
    ## [2,]  0.5 -0.25
    ```

]

.pull-right[
- A matrix multiplied by its inverse gives the identity matrix.
    
    ```r
    A1 %*% solve(A1)
    ```
    
    ```
    ##      [,1] [,2]
    ## [1,]    1    0
    ## [2,]    0    1
    ```
    
- `$3\times 3$` example:
    
    ```r
    E
    ```
    
    ```
    ##      [,1] [,2] [,3]
    ## [1,]    7    2    1
    ## [2,]    9    5    3
    ## [3,]    4    6    8
    ```
    
    ```r
    solve(E) %>% round(3)
    ```
    
    ```
    ##        [,1]   [,2]   [,3]
    ## [1,]  0.324 -0.147  0.015
    ## [2,] -0.882  0.765 -0.176
    ## [3,]  0.500 -0.500  0.250
    ```

]
---

### Patterned Matrices

.pull-left[
- Create a diagonal matrix:
    
    ```r
    diag(1,4,4)
    ```
    
    ```
    ##      [,1] [,2] [,3] [,4]
    ## [1,]    1    0    0    0
    ## [2,]    0    1    0    0
    ## [3,]    0    0    1    0
    ## [4,]    0    0    0    1
    ```
]

.pull-right[
- Extract the diagonal elements of a matrix:
    
    ```r
    diag(A1)
    ```
    
    ```
    ## [1] 1 6
    ```
]

***

- More complex patterned matrices can be created using the methods we covered, and these are often used within financial econometric analyses. For example:
    
    ```r
    diag(rep(c(1,2),c(6,6)))
    ```
    
    ```
    ##       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
    ##  [1,]    1    0    0    0    0    0    0    0    0     0     0     0
    ##  [2,]    0    1    0    0    0    0    0    0    0     0     0     0
    ##  [3,]    0    0    1    0    0    0    0    0    0     0     0     0
    ##  [4,]    0    0    0    1    0    0    0    0    0     0     0     0
    ##  [5,]    0    0    0    0    1    0    0    0    0     0     0     0
    ##  [6,]    0    0    0    0    0    1    0    0    0     0     0     0
    ##  [7,]    0    0    0    0    0    0    2    0    0     0     0     0
    ##  [8,]    0    0    0    0    0    0    0    2    0     0     0     0
    ##  [9,]    0    0    0    0    0    0    0    0    2     0     0     0
    ## [10,]    0    0    0    0    0    0    0    0    0     2     0     0
    ## [11,]    0    0    0    0    0    0    0    0    0     0     2     0
    ## [12,]    0    0    0    0    0    0    0    0    0     0     0     2
    ```

---

## Linear regression model

- The linear regression model in matrix form. Consider as the unknown "true" model that we want to know about.

$$
    y = X\beta + \epsilon
    $$

- Where `$y$` is the dependent variable, `$X$` contains explanatory variables, `$\beta$` contains the parameters to be estimated and `$\epsilon$` is a stochastic disturbance or random error term.

<BR/>

`$$\underset{n\times 1}y = \begin{bmatrix}
Y_1\\
Y_2\\
\vdots\\
Y_n
\end{bmatrix}
,\;
\underset{n\times (k+1)}X = \begin{bmatrix}
1 & X_{11} & X_{12} & \dots & X_{k1}\\
1 & X_{21} & X_{22} & \dots & X_{k2}\\
\vdots\\
1 & X_{n1} & X_{n2} & \dots & X_{kn}
\end{bmatrix}
,\;
\underset{(k+1)\times 1}\beta = \begin{bmatrix}
\beta_1\\
\beta_2\\
\vdots\\
\beta_k
\end{bmatrix}
,\;
\underset{n\times 1}\epsilon = \begin{bmatrix}
\epsilon_1\\
\epsilon_2\\
\vdots\\
\epsilon_n
\end{bmatrix}$$`
<BR/>

- We have `$n$` observations of the dependent variable, `$n$` observations of `$k$` explanatory variables, an intercept term (the column of ones in `$X$`), and `$k+1$` parameters (the intercept plus the coefficients of our `$k$` explanatory variables).

---

## Linear regression model - example

- Assume a model with an intercept, two explanatory variables and we have 30 observations.

`$$y_i = \beta_1 + \beta_2 X_{i1} + \beta_3 X_{i2} + \epsilon_i \quad i=1,\dots,30$$`
    
- Written in matrix form:

$$
    y = X\beta + \epsilon
    $$

<BR/>

`$$\underset{30\times 1}y = \begin{bmatrix}
Y_1\\
Y_2\\
\vdots\\
Y_{30}
\end{bmatrix}
,\quad
\underset{30\times 3}X = \begin{bmatrix}
1 & X_{11} & X_{12}\\
1 & X_{21} & X_{22}\\
\vdots\\
1 & X_{301} & X_{302}
\end{bmatrix}
,\quad
\underset{3\times 1}\beta = \begin{bmatrix}
\beta_1\\
\beta_2\\
\beta_3
\end{bmatrix}
,\quad
\underset{30\times 1}\epsilon = \begin{bmatrix}
\epsilon_1\\
\epsilon_2\\
\vdots\\
\epsilon_{30}
\end{bmatrix}$$`

---

## Ordinary Least Squares Estimation

- Our aim is to generate the best possible estimates of the parameters, `$\beta$`.

- Our estimate of `$\beta$` is denoted `$\hat\beta$` and the estimate of the error term is the residual `$e$`. The residual contains what our model does not explain about `$y$`.

`$$e=y-X\hat\beta$$`

- Ordinary Least Squares (OLS) estimation simply says select the `$\hat\beta$` that minimises what our model (in particular, `$X\hat\beta$`) does not explain about our dependent variable `$y$`.

- We do this by finding the `$\hat\beta$` that minimises the sum of the squared residuals, `$e^\prime e$`.
    
    - Note that the *fitted values* of our regression are `$\hat y = X\hat\beta$`.
    
    - Next we derive the Ordinary Least Squares Estimator (OLSE). 
    
---

## Ordinary Least Squares Estimation

- Sum of the squared residuals, `$e^{\prime}e$`:

`$$\begin{align}\underset{1\times n}{e^{\prime}} \times \underset{n\times 1}{e} &=
\begin{bmatrix}
e_1\: e_2\: \dots\: e_n
\end{bmatrix}
\begin{bmatrix}
e_1\\
e_2\\
\dots\\
e_n
\end{bmatrix}\\[9pt]
&=\underset{1\times 1}{\begin{bmatrix}
e_1\times e_1\, +\, e_2\times e_2\, +\, \dots,\, +\, e_n\times e_n
\end{bmatrix}}\end{align}$$`

- Write the sum of the squared residuals in terms of our data and the parameter estimates.

- Note that the transpose of a scalar is a scalar, so `$\hat{\beta^{\prime}}X^{\prime}y=y^{\prime}X\hat\beta$`.

`$$\begin{align}e^{\prime}e &= {(y-X\hat\beta)}^{\prime}(y-X\hat\beta)\\
&= y^{\prime}y - \hat{\beta^{\prime}}X^{\prime}y - y^{\prime}X\hat\beta +\hat{\beta^{\prime}}X^{\prime}X\hat\beta\\
&= y^{\prime}y - 2\hat{\beta^{\prime}}X^{\prime}y + \hat{\beta^{\prime}}X^{\prime}X\hat\beta\end{align}$$`

---

## Ordinary Least Squares Estimation

- To find the `$\hat\beta$` that minimises the sum of the squared residuals, differentiate `$e'e$` with respect to `$\hat\beta$` and set equal to zero.

`$$\begin{align}\frac{\partial e^{\prime}e}{\partial \hat\beta} &= -2X^{\prime}y + 2X^{\prime}X\hat\beta = 0
\end{align}$$`

`$$X^{\prime}X\hat\beta = X^{\prime}y$$`

- Pre-multiply each side by `${(X^{\prime}X)}^{-1}$`:

`$${(X^{\prime}X)}^{-1}X^{\prime}X\hat\beta={(X^{\prime}X)}^{-1}X^{\prime}y$$`

- The OLSE of `$\beta$` is `$\hat\beta$`:

`$$\hat\beta={(X^{\prime}X)}^{-1}X^{\prime}y$$`

---

## Finance terms

- We will analyse stock markets in this course.

- A stock represents<sup>2</sup>

- [Investopedia](https://www.investopedia.com/) is a good place to look up finance terms.

.footnote[[2] Text associated with footnote 1.]

---

### Nikkei NEEDS and Financial Quest

---

## References

Ang, C. S. (2021). _Analyzing Financial Data and Implementing Financial
Models Using R_. Second. Springer Texts in Business and Economics.
Cham: Springer International Publishing. ISBN: 978-3-030-64154-2. DOI:
[10.1007/978-3-030-64155-9](https://doi.org/10.1007%2F978-3-030-64155-9).
URL:
[https://link.springer.com/10.1007/978-3-030-64155-9](https://link.springer.com/10.1007/978-3-030-64155-9).

Bennett, M. J. and D. L. Hugen (2016). _Financial Analytics with R:
Building a Laptop Laboratory for Data Science_. Cambridge University
Press. ISBN: 9781107150751. DOI:
[https://doi.org/10.1017/CBO9781316584460](https://doi.org/https%3A%2F%2Fdoi.org%2F10.1017%2FCBO9781316584460).

Kleiber, C. and A. Zeileis (2008). _Applied Econometrics with R_. 1st.
New York: Springer US. ISBN: 978-0-387-77316-2. DOI:
[10.1007/978-0-387-77318-6](https://doi.org/10.1007%2F978-0-387-77318-6).
eprint: arXiv:1011.1669v3. URL:
[http://link.springer.com/10.1007/978-0-387-77318-6
http://www.springer.com/gp/book/9780387773162](http://link.springer.com/10.1007/978-0-387-77318-6
http://www.springer.com/gp/book/9780387773162).

Matloff, N. (2011). _The Art of R Programming: A Tour of Statistical
Software Design_. No Starch Press. ISBN: 9781593273842. URL:
[https://www.nostarch.com/artofr.htm](https://www.nostarch.com/artofr.htm).

Pfaff, B. (2008). _Analysis of Integrated and Cointegrated Time Series
with R_. Second. New York: Springer-Verlag. ISBN: 978-0-387-75966-1.
URL: [papers2://publication/uuid/2D6BDE05-D712-4D3C-99CD-72EADA215C83
http://www.springer.com/gp/book/9780387759661](papers2://publication/uuid/2D6BDE05-D712-4D3C-99CD-72EADA215C83
http://www.springer.com/gp/book/9780387759661).

---

Tsay, R. S. (2014). _Multivariate time series analysis : with R and
financial applications_. Wiley. ISBN: 9781118617908.

---

## Appendix

- Locale issues:

```r
Sys.setlocale("LC_ALL", "en_US.UTF-8")
```

---

### Extra screen rec.

- [Amazon](https://www.amazon.co.jp/dp/B07ZNTHZHZ)